Hermitian function

In mathematical analysis, a Hermitian function is a complex function with the property that its complex conjugate is equal to the original function with the variable changed in sign:


 * $$f^*(x) = f(-x)$$

(where the $$^*$$ indicates the complex conjugate) for all $$x$$ in the domain of $$f$$. In physics, this property is referred to as PT symmetry.

This definition extends also to functions of two or more variables, e.g., in the case that $$f$$ is a function of two variables it is Hermitian if


 * $$f^*(x_1, x_2) = f(-x_1, -x_2)$$

for all pairs $$(x_1, x_2)$$ in the domain of $$f$$.

From this definition it follows immediately that: $$f$$ is a Hermitian function if and only if


 * the real part of $$f$$ is an even function,
 * the imaginary part of $$f$$ is an odd function.

Motivation
Hermitian functions appear frequently in mathematics, physics, and signal processing. For example, the following two statements follow from basic properties of the Fourier transform:

Since the Fourier transform of a real signal is guaranteed to be Hermitian, it can be compressed using the Hermitian even/odd symmetry. This, for example, allows the discrete Fourier transform of a signal (which is in general complex) to be stored in the same space as the original real signal.
 * The function $$f$$ is real-valued if and only if the Fourier transform of $$f$$ is Hermitian.
 * The function $$f$$ is Hermitian if and only if the Fourier transform of $$f$$ is real-valued.


 * If f is Hermitian, then $$f \star g = f*g$$.

Where the $$ \star $$ is cross-correlation, and $$ * $$ is convolution.


 * If both f and g are Hermitian, then $$f \star g = g \star f$$.